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Prove that the angle bisectors of alternate interior angles are parallel. Given: AB II CD, CB is the angle bisector of

Prove that the angle bisectors of alternate interior angles are parallel. Given: AB II CD, CB is the angle bisector of

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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## Proof: Angle Bisectors of Alternate Interior Angles Are Parallel ### Given: 1. Lines \( \overline{AB} \parallel \overline{CD} \) 2. Line \( \overline{CB} \) is the angle bisector of \( \angle FCA \) 3. Line \( \overline{AD} \) is the angle bisector of \( \angle EAC \) ### To Prove: \( \overline{CB} \parallel \overline{AD} \) ### Diagram Explanation: The diagram provided demonstrates the geometric configuration of the given conditions: 1. **Parallel Lines** \( \overline{AB} \) and \( \overline{CD} \) are shown as black lines intersected by another black line that is cutting across both, creating angles. 2. **Angle Bisectors**: - The red-marked region at point \( C \) indicates \( \overline{CB} \), which bisects \( \angle FCA \). - The orange-marked region at point \( A \) signifies \( \overline{AD} \), which bisects \( \angle EAC \). ### Proof Steps: This proof involves geometric concepts and properties of parallel lines and angle bisectors. Follow the steps below: 1. **Identify the Alternate Interior Angles**: - Since \( \overline{AB} \parallel \overline{CD} \) and they are intersected by the black line, \( \angle FCA \) and \( \angle EAC \) are formed as alternate interior angles. 2. **Apply the Angle Bisector Theorem**: - By definition, an angle bisector divides an angle into two equal parts. - Therefore, \( \overline{CB} \) bisects \( \angle FCA \) and \( \overline{AD} \) bisects \( \angle EAC \). 3. **Determine the Parallelism of Bisectors**: - Given the alternate interior angles are equal, their bisectors will maintain the same relative direction. - Since parallel lines maintain equivalent alternate interior angles on both sides of the transversal, and each angle's halves specified by their bisectors must also align in a parallel manner. Hence, the angle bisectors \( \overline{CB} \) and \( \overline{AD} \) are shown to be parallel by their respective angle
Transcribed Image Text:## Proof: Angle Bisectors of Alternate Interior Angles Are Parallel ### Given: 1. Lines \( \overline{AB} \parallel \overline{CD} \) 2. Line \( \overline{CB} \) is the angle bisector of \( \angle FCA \) 3. Line \( \overline{AD} \) is the angle bisector of \( \angle EAC \) ### To Prove: \( \overline{CB} \parallel \overline{AD} \) ### Diagram Explanation: The diagram provided demonstrates the geometric configuration of the given conditions: 1. **Parallel Lines** \( \overline{AB} \) and \( \overline{CD} \) are shown as black lines intersected by another black line that is cutting across both, creating angles. 2. **Angle Bisectors**: - The red-marked region at point \( C \) indicates \( \overline{CB} \), which bisects \( \angle FCA \). - The orange-marked region at point \( A \) signifies \( \overline{AD} \), which bisects \( \angle EAC \). ### Proof Steps: This proof involves geometric concepts and properties of parallel lines and angle bisectors. Follow the steps below: 1. **Identify the Alternate Interior Angles**: - Since \( \overline{AB} \parallel \overline{CD} \) and they are intersected by the black line, \( \angle FCA \) and \( \angle EAC \) are formed as alternate interior angles. 2. **Apply the Angle Bisector Theorem**: - By definition, an angle bisector divides an angle into two equal parts. - Therefore, \( \overline{CB} \) bisects \( \angle FCA \) and \( \overline{AD} \) bisects \( \angle EAC \). 3. **Determine the Parallelism of Bisectors**: - Given the alternate interior angles are equal, their bisectors will maintain the same relative direction. - Since parallel lines maintain equivalent alternate interior angles on both sides of the transversal, and each angle's halves specified by their bisectors must also align in a parallel manner. Hence, the angle bisectors \( \overline{CB} \) and \( \overline{AD} \) are shown to be parallel by their respective angle
### Problem Submission Guidelines To receive full credit for your answer, ensure that your submission includes the following components: - **An Informal Paragraph Proof of the Problem** - This should be a clear and concise explanation of your reasoning in paragraph form, demonstrating your understanding of the problem and how to solve it. - **A Formal Two-Column Proof of the Problem** - This should be properly structured, with one column for statements and another for the corresponding reasons. Your reasons must include definitions, theorems, postulates, and/or properties of congruence or equality. Before submitting your response, make sure to attach any necessary files.
Transcribed Image Text:### Problem Submission Guidelines To receive full credit for your answer, ensure that your submission includes the following components: - **An Informal Paragraph Proof of the Problem** - This should be a clear and concise explanation of your reasoning in paragraph form, demonstrating your understanding of the problem and how to solve it. - **A Formal Two-Column Proof of the Problem** - This should be properly structured, with one column for statements and another for the corresponding reasons. Your reasons must include definitions, theorems, postulates, and/or properties of congruence or equality. Before submitting your response, make sure to attach any necessary files.
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